Rectified great stellated hecatonicosachoron

Rectified great stellated hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Ragishi
Coxeter diagram	o5/2x3o5o ()
Elements
Cells	120 icosahedra, 120 great icosidodecahedra
Faces	2400 triangles, 720 pentagrams
Edges	3600
Vertices	720
Vertex figure	Semi-uniform Pentagonal prism, edge lengths 1 (base) and (√5–1)/2 (side)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5-\sqrt5}{2}} ≈ 1.17557$
Hypervolume	$5\frac{205-63\sqrt5}{4} ≈ 80.15965$
Dichoral angles	Gid–5/2–gid: 144°
	Ike–3–gid: 120°
Central density	20
Number of external pieces	5160
Level of complexity	20
Related polytopes
Army	Rox, edge length $\frac{3-\sqrt5}{2}$
Regiment	Ragishi
Conjugate	Rectified grand hecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	43200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The rectified great stellated hecatonicosachoron, or ragishi, is a nonconvex uniform polychoron that consists of 120 icosahedra and 120 great icosidodecahedra. Two icosahedra and five great icosidodecahedra join at each pentagonal prismatic vertex. As the name suggests, it can be obtained by rectifying the great stellated hecatonicosachoron.

Gallery

Vertex coordinates

The vertices of a rectified great stellated hecatonicosachoron of edge length 1 are given by all permutations of:

• $\left(0,\,0,\,±1,\,±\frac{\sqrt5-1}{2}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{4}\right),$

along with even permutations of:

• $\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac12,\,±\frac{5-\sqrt5}{4}\right),$
• $\left(0,\,±\frac{\sqrt5-1}{4},\,±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5}{2}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac12,\,±\frac{\sqrt5-1}{2},\,±\frac{\sqrt5-1}{4}\right),$
• $\left(±\frac12,\,±\frac{\sqrt5-1}{4},\,±1,\,±\frac{3-\sqrt5}{4}\right).$

Related polychora

The rectified great stellated hecatonicosachoron is the colonel of a regiment with 15 members. Of these, one other besides the colonel itself is Wythoffian (the rectified grand stellated hecatonicosachoron), two are hemi-Wythoffian (the small pentagrammal antiprismatoverted dishecatonicosachoron and pentagonal retroprismatoverted hexacosihecatonicosachoron), and one is noble (the great retropental hecatonicosachoron).

It has the same circumradius as the hexagonal-decagrammic duoprism.

Uniform polychoron compounds composed of rectified great stellated hecatonicosachora include:

• Rectified great stellated diacositetracontachoron (2)
• Rectiied great stellated chirohexacosichoron (5)
• Rectified great stellated dishexacosichoron (10)

External links

• Bowers, Jonathan. "Category 5: Pentagonal Rectates" (#103).

• Klitzing, Richard. "ragishi".